Self-Orthogonal Mendelsohn Triple Systems

Abstract

A Mendelsohn triple system of order v , briefly MTS( v ), is a pair ( X , B ) where X is a v -set (of points ) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B . The cyclic triple ( a, b, c ) contains the ordered pairs ( a, b ), ( b, c ), and ( c, a ). An MTS( v ) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order v . An MTS( v ) is called self-orthogonal , denoted briefly by SOMTS( v ), if its associated semisymmetric Latin square is self-orthogonal. It is well known that an MTS( v ) exists if and only if v ≡ 0 or 1 (mod 3) except v ≠ 6. It is also known that a SOMTS( v ) exists for all v ≡ 1 (mod 3) except v = 10 and that a SOMTS( v ) does not exist for v = 3, 6, 9, and 12. In this paper it is shown that a SOMTS( v ) exists for all v ⩾ 15, where v ≡ 0 (mod 3), except possibly for v = 18.